If $\mathrm{m} \overparen{T A}=42^{\circ}$ and $\mathrm{m} \overparen{M I}=130^{\circ}$, find $\mathrm{m} \angle C$.
Answer
Final Answer: The measure of angle C is \( \boxed{86} \) degrees.
Step 1 ：Given that the measure of arc TA is 42 degrees and the measure of arc MI is 130 degrees.
Step 2 ：In a circle, the measure of an angle formed by two chords is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Step 3 ：So, the measure of angle C is half the sum of the measures of arc TA and arc MI.
Step 4 ：Let's calculate this: \( m_{TA} = 42 \), \( m_{MI} = 130 \)
Step 5 ：Using the formula, we get \( m_C = \frac{m_{TA} + m_{MI}}{2} = \frac{42 + 130}{2} = 86.0 \)
Step 6 ：Final Answer: The measure of angle C is \( \boxed{86} \) degrees.
